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Entropy

On the Optimization and Self-Tensoring Construction of Nonlinear Entanglement Witnesses

Juan Yu, Lei Li, Ming Li, Shu‐Qian Shen

Published May 28, 2026• 0 citations
Publisher

Abstract

Nonlinear entanglement witnesses constructed from multiple linear entanglement witnesses and multiple copies of quantum states have recently been proposed as a powerful tool for entanglement detection. In this work, we show, via an explicit counterexample, that the fineness of linear witnesses generally fails to transfer to their tensor-product nonlinear counterparts. For the canonical family of nonlinear witnesses in the form of (αI-L)⊗(βI-T), we rigorously prove that the optimal nonlinear witness is uniquely attained with weakly optimal parameters of α=λmax(L) and β=λmax(T). Meanwhile, we analytically demonstrate that the self-tensor products of two representative linear witnesses fail to detect any entangled state. The question of whether a nonlinear entanglement witness capable of detecting entanglement can be constructed by tensoring a linear witness with itself remains open.

DOI
10.3390/e28060608

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Nonlinear entanglement witnesses constructed from multiple linear entanglement witnesses and multiple copies of quantum states have recently been proposed as a powerful tool for entanglement detection. In this work, we show, via an explicit counterexample, that the fineness of linear witnesses generally fails to transfer to their tensor-product nonlinear counterparts. For the canonical family of nonlinear witnesses in the form of (αI-L)⊗(βI-T), we rigorously prove that the optimal nonlinear witness is uniquely attained with weakly optimal parameters of α=λmax(L) and β=λmax(T). Meanwhile, we analytically demonstrate that the self-tensor products of two representative linear witnesses fail to detect any entangled state. The question of whether a nonlinear entanglement witness capable of detecting entanglement can be constructed by tensoring a linear witness with itself remains open.
We construct linear and nonlinear entanglement witnesses, by tensoring and partial tracing existing states and witnesses. We show that little shared quantum resources allow one to employ decomposable witnesses to obtain larger ones detecting locally undistillable states and find analytic witnesses for multipartite entangled states that are undistillable across all bipartitions. Crucially, we show how to efficiently optimize multicopy witnesses to desired states, which is a current challenge. As an example of both theoretical and experimental interest, existing trace polynomial witnesses are significantly improved while preserving their symmetries and implementability with randomized measurements. In addition to detecting entanglement, we also find new witnesses detecting k-copy distillability. A recipe for the single-copy case is shown to be effective for generic and Werner states.
Entanglement detection is one of the important problems in quantum information theory. To deal with this problem, many entanglement detection criteria have been proposed. Among the proposed criteria, the detection of entanglement through witness operator (also known as linear entanglement witness (LEW) operator) may be considered as the most practical. Although the witness operator approach to detect entanglement is experimentally friendly, the construction of these operators is not a very simple task. Even if we are able to construct a LEW operator, our problem is not solved as it may either detect a few entangled states or not a single entangled state from a given family of entangled states. Thus, we need a constructive approach in order to tackle this type of problem. In this work, we provide a few constructions of the non-linear entanglement witnesses (NLEW) for d1⊗d2d_1\otimes d_2d1​⊗d2​ dimensional system from any linear entanglement witness (LEW) operator. The advantage of these constructions is that, if a LEW is unable to detect any particular entangled state described by the density operator ρentρ^{ent}ρent then our construction of NLEW may detect the same entangled state ρentρ^{ent}ρent. Further, we have constructed NLEW operator that may detect not only a class of bipartite negative partial transpose entangled state (NPTES), but also positive partial transpose entangled state (PPTES). Moreover, we have shown that the constructed NLEW operators may be decomposed in terms of the tensor product of local observables and hence may be realizable in an experiment.
We present a nonlinear entanglement detection strategy that detects entanglement that the linear detection strategy fails. We show that when the nonlinear entanglement detection strategy fails to detect the entanglement of an entangled state with two copies, it may succeed with three or more copies. Based on our strategy, a witness combined with a suitable quantum mechanical observable may detect the entanglement that cannot be detected by the witness alone. Moreover, our strategy can also be applied to detect multipartite entanglement by using the witnesses for bipartite systems, as well as to entanglement concentrations.
Existing methods for deciding whether a bipartite quantum state is separable or entangled typically fall into one of two categories: they are either complete but require access to an explicit density matrix followed by numerical optimization, or they can be evaluated directly by measuring the quantum system but are incomplete, in the sense that they cannot detect all forms of entanglement. In this work, we overcome both limitations in a unified framework. First, we bypass numerical optimization by deriving separability criteria in the form of universal bounds on tensor powers of separable states. We prove that these bounds are complete: every entangled state violates them for sufficiently large tensor powers. Second, we explicitly construct a corresponding complete family of nonlinear entanglement witnesses, which can detect all forms of entanglement without requiring an explicit density matrix. The witnesses we construct are moreover basis-independent, in the sense that they are invariant under conjugation by local unitaries. Altogether, our results expand the toolbox for entanglement detection in arbitrary local dimensions in a manifestly invariant way.
Verification of entanglement is an important tool to characterize sources and devices for use in quantum computing and communication applications. In a vast majority of experiments entanglement witnesses (EWs) are used in order to prove the presence of entanglement in a quantum state. EWs can be constructed from available measurement results and do not require a reconstruction of the whole density matrix (full tomography), which is especially valuable for high-dimensional systems. We provide a method to construct accessible nonlinear EWs, which incorporate two important properties. First, they improve on linear EWs in the sense that each nonlinear EW detects more entangled states than its linear counterpart and therefore allows the verification of entanglement without critical dependence on having found the ``right'' linear witness. Second, they can be evaluated using exactly the same data as for the evaluation of the original linear witness. This allows a reanalysis of published experimental data to strengthen statements about entanglement verification without the requirement to perform additional measurements. These particular properties make the accessible nonlinear EWs attractive for the implementations in current experiments, for they can also enhance the statistical significance of the entanglement verification.
Entanglement witnesses (EWs) are fundamental tools for detecting entanglement. However, traditional linear witnesses often fail to identify most of the entangled states. In this work, we construct a family of nonlinear entanglement witnesses by taking the envelope of linear witnesses defined over continuous families of pure bipartite states with fixed Schmidt bases. This procedure effectively ``bends'' the hyperplanes associated with linear witnesses into curved hypersurfaces, thereby extending the region of detectable entangled states. The resulting conditions can be expressed in terms of the positive semidefiniteness of a family of matrices, whose principal minors define a hierarchy of increasingly sensitive detection criteria. We show that this construction is not limited to the transposition map and generalizes naturally to arbitrary positive but not completely positive maps, leading to nonlinear analogs of general entanglement witnesses. We emphasize that the required measurements remain experimentally accessible, as the nonlinear criteria are still formulated in terms of expectation values over local operator bases. Through both analytical and numerical examples, we demonstrate that the proposed nonlinear witnesses outperform their linear counterparts in detecting entangled states which may evade individual linear EWs in the construction. This approach offers a practical and conceptually elegant enhancement to entanglement detection in finite-dimensional systems.